# Quantifying channels output similarity with applications to quantum control

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## Abstract

In this work, we aim at quantifying quantum channel output similarity. In order to achieve this, we introduce the notion of quantum channel superfidelity, which gives us an upper bound on the quantum channel fidelity. This quantity is expressed in a clear form using the Kraus representation of a quantum channel. As examples, we show potential applications of this quantity in the quantum control field.

## Keywords

Quantum control Fidelity Superfidelity## 1 Introduction

Recent applications of quantum mechanics are based on processing and transferring information encoded in quantum states. The full description of quantum information processing procedures is given in terms of quantum channels, i.e. completely positive, trace- preserving maps on the set of quantum states.

In many areas of quantum information processing, one needs to quantify the difference between ideal quantum procedure and the procedure which is performed in the laboratory. This is especially true in the situation when one deals with imperfections during the realization of experiments. These imperfections can be countered, in a quantum control setup, using various techniques, such us dynamical decoupling [1, 2, 3, 4], sliding mode control [5] and risk sensitive quantum control [6, 7]. A different approach is to model the particular setup and optimize control pulses for a specific task in a specific setup [8, 9, 10, 11]. In particular, the problem of quantifying the distance between quantum channels was studied in the context of channel distinguishability.

One possible approach to quantifying the distance between two quantum channels is to consider the fidelity between Choi–Jamiołkowski states corresponding to quantum channels [12]. Another approach could involve the diamond norm [13] of quantum channels. We propose an approach which focuses on the outputs of quantum channels.

The main aim of this paper is to provide a succinct expression for the channel output similarity. As a measure of similarity, we will consider the superfidelity function and define channel superfidelity. Then we will show examples of application of our results to various pairs of quantum channels. In the final part of the paper, we will study the impact of Hamiltonian errors on the channel superfidelity. First, we will consider a single qubit at a finite temperature, and next we will move to an extended quantum control example.

## 2 Preliminaries

Henceforth, we will denote the set of linear operators, transforming vectors from a finite-dimensional Hilbert space \(\mathcal {X}\) to another finite-dimensional Hilbert space \(\mathcal {Y}\) by \(\mathcal {L}(\mathcal {X}, \mathcal {Y})\). We put \(\mathcal {L}(\mathcal {X}) = \mathcal {L}(\mathcal {X}, \mathcal {X})\). By \(\mathcal {U}(\mathcal {X})\), we will denote the set of unitary operators on \(\mathcal {X}\). Given an operator \(A \in \mathcal {L}(\mathcal {X}, \mathcal {Y})\), we denote by \(\Vert A\Vert _p\) its Schatten p-norm. By \(\bar{A}\), we will denote the element-wise complex conjugation of *A*.

### 2.1 Quantum states and channels

First, we introduce two basic notions: density operators and superoperatros:

### **Definition 1**

We call an operator \(\rho \in \mathcal {L}(\mathcal {X})\) a density operator iff \(\rho \ge 0\) and \(\mathrm {Tr}\rho = 1\). We denote the set of all density operators on \(\mathcal {X}\) by \(\mathcal {D}(\mathcal {X})\).

From this follows that \(\rho \) is in the form \(\rho =\sum _j \lambda _j \left| \lambda _j\right\rangle \left\langle \lambda _j\right| \), where \(\lambda _j\) and \(\left| \lambda _j\right\rangle \) denote the *j*th eigenvalue and eigenvector of \(\rho \), respectively.

### **Definition 2**

Now we define the tensor product of superoperators

### **Definition 3**

In the most general case, the evolution of a quantum system can be described using the notion of a *quantum channel* [14, 15, 16].

### **Definition 4**

- 1.
\(\varPhi \) is trace preserving, i.e. \(\forall {A \in \mathcal {L}(\mathcal {X})} \quad \mathrm {Tr}(\varPhi (A))=\mathrm {Tr}(A)\),

- 2.\(\varPhi \) is completely positive, that is for every finite-dimensional Hilbert space \(\mathcal {Z}\) the product of \(\varPhi \) and identity mapping on \(\mathcal {L}(\mathcal {Z})\) is a non-negativity-preserving operation, i.e.$$\begin{aligned} \forall {\mathcal {Z}} \ \forall {A \in \mathcal {L}(\mathcal {X}\otimes \mathcal {Z})} \quad {A \ge 0} \Rightarrow \varPhi \otimes \mathbbm {1}_{\mathcal {L}(\mathcal {Z})}(A) \ge 0. \end{aligned}$$(5)

Many different representations of quantum channels can be chosen, depending on the application. In this paper, we will use only the Kraus representation.

### **Definition 5**

### 2.2 Superfidelity

In this section, we introduce the *superfidelity*, along with its properties

### **Definition 6**

The superfidelity is an upper bound for the fidelity function [14, 17].

- 1.
Bounds: \(0 \le G(\rho _1, \rho _2) \le 1\).

- 2.
Symmetry: \(G(\rho _1, \rho _2) = G(\rho _2, \rho _1)\).

- 3.
Unitary invariance: \(G(\rho _1, \rho _2) = G(U \rho _1 U^\dagger , U \rho _2 U^\dagger )\), where \(U \in \mathcal {U}(\mathcal {X})\).

- 4.Joint concavity [18]:for \(p \in [0, 1]\).$$\begin{aligned} G (p \rho _1 + (1-p)\rho _2, p \rho _3 + (1-p)\rho _4) \le pG(\rho _1, \rho _3) + (1-p) G(\rho _2, \rho _4) \end{aligned}$$(9)
- 5.Supermultiplicavity:$$\begin{aligned} G(\rho _1 \otimes \rho _2, \rho _3 \otimes \rho _4) \ge G(\rho _1, \rho _3) G(\rho _2, \rho _4). \end{aligned}$$(10)
- 6.Bound for trace distance [19]$$\begin{aligned} \frac{1}{2} \Vert \rho _1 - \rho _2 \Vert _1 \ge 1 - G(\rho _1, \rho _2). \end{aligned}$$(11)

### 2.3 Supporting definitions

In this section, we define additional operations used in our proof. We begin with the *partial trace*

### **Definition 7**

*A*,

*B*the partial trace is a linear mapping defined as:

We will also need the notion of conjugate superoperator

### **Definition 8**

Note that the conjugate to completely positive superoperator is completely positive, but is not necessarily trace preserving

Next, we will define a reshaping operation, which preservers the lexicographical order and its inverse.

### **Definition 9**

We introduce the inverse of the \(\mathrm {res}(\cdot )\)

### **Definition 10**

### *Remark 1*

### *Remark 2*

Next, we introduce the *purification* of quantum states:

### **Definition 11**

### **Theorem 1**

### 2.4 Quantum channel fidelity

First, we introduce the *fidelity* and *channel fidelity* [12]

### **Definition 12**

### **Definition 13**

## 3 Our results

In this section, we present our main theorem and its proof. In the second subsection, we present a quantum circuit that allows one to measure the quantum channel superfidelity without performing full state tomography.

### 3.1 Theorem and proof

### **Definition 14**

The channel superfidelity \(G_\mathrm {ch}(\varPhi , \varPsi ; \sigma )\) places a lower bound on the output superfidelity of two quantum channels in the case of the same input states. Henceforth, where unambiguous, we will write the channel superfidelity as \(G_\mathrm {ch}\).

### **Theorem 2**

### *Proof*

The following simple corollaries are easily derived from Theorem 2.

### **Corollary 1**

### *Proof*

Assume \(\varPhi \) has the Kraus form \(\{K_i\}_i\). Substituting the identity for \(L_j\) in Eq. (27), we recover Eq. (25) which completes the proof. \(\square \)

### **Corollary 2**

### *Proof*

If \(\varPhi \) is a unitary channel, then the second term in Eq. (28) vanishes. Let us assume that \(\varPsi \) has a Kraus form \(\{L_j\}_j\). We get \(G_\mathrm {ch} = \sum _j |\mathrm {Tr}\sigma U^\dagger L_j|^2\).

The Kraus form of the channel \(\varPsi '\) is given by the set \(\{ U^\dagger L_j: K_j \in \mathcal {L}(\mathcal {X}) \}\). Using this in Eq. (25), we get \(F_\mathrm {ch}(\varPsi '; \sigma ) = \sum _j|\mathrm {Tr}\sigma U^\dagger L_j|^2\). This completes the proof. \(\square \)

### **Corollary 3**

### *Proof*

*T*and \(T_\mathrm {ch}\), respectively. Let us assume that channels \(\varPhi \) and \(\varPsi \) have Kraus forms \(\{K_i\}_i\) and \(\{L_j \}_j\), respectively. We get:

### 3.2 Quantum circuit for measuring channel superfidelity

Note that this approach is far simpler, compared to estimating the channel fidelity which would require us to perform full state tomography. Furthermore, analytical calculations involving fidelity get cumbersome quickly, as it requires calculating expressions of the form \(\Vert \sqrt{\varPhi (\sigma )} \sqrt{\varPsi (\sigma )}\Vert _1\).

## 4 Simple examples

In this section, we provide a number of examples of the application of Theorem 2.

### 4.1 Erasure channel

### **Definition 15**

*A*in \(\mathcal {L}(\mathcal {X})\). The Kraus form of this channel is given by the set \(\{ K_{ij}: K_{ij} = \sqrt{\lambda _i} \left| \lambda _i\right\rangle \left\langle j\right| \}_{ij}\), \(\lambda _i\) and \(\left| \lambda _i\right\rangle \) denote the

*i*th eigenvalue and the corresponding eigenvector of \(\xi \).

### 4.2 Sensitivity to channel error

*H*of the operator \(K_j\sigma K_i^\dagger \).

## 5 Sensitivity to Hamiltonian parameters

In this section, we will show how the channel superfidelity is affected by errors in the system Hamiltonian parameters. First, we will show analytical results for a single qubit system at a finite temperature. Next, we show numerical results for a simple, three-qubit spin chain.

### 5.1 Single qubit at a finite temperature

*T*, Eq. (42) may be rewritten as

*reshuffle*operation on matrix

*M*[14]. Now, it is simple to find the Kraus form of the channel \(\varPhi _T^\epsilon \). The Kraus operators are related to the eigenvalues \(\lambda _i\) and eigenvectors \(\left| \lambda _i\right\rangle \) of \(D_{\varPhi _T^\epsilon }\) in the following manner:

*T*and second when \(\epsilon T = \frac{\pi }{2}\). As we are mainly interested in small values of \(\epsilon \), we expand \(\cos \epsilon T\) up to the second term in the Taylor series. We get:

### 5.2 Quantum control example

*i*. We set the control Hamiltonian \(H_\mathrm {c}\) to:

*s*. We have conducted 100 simulations for each value of

*s*. As expected, the quantum channel superfidelity decreases slowly for low values of

*s*. After a certain value, the decrease becomes rapid. As values of

*s*increase, the minimum and maximum achieved fidelity diverge rapidly. This is represented by the shaded area in Fig. 2. We can approximate the average value of the channel fidelity as \(\langle G_\mathrm {ch} \rangle \approx 1 - c s ^2\). Fitting this function to the curve shown in Fig. 2b gives a relative error which is less then 0.5 %.

## 6 Conclusions

We have studied the superfidelity of a quantum channel. This quantity allows us to provide an upper bound on the fidelity of the output of two quantum channels. We shown an example of application of this quantity to a unitary and an erasure channel. The obtained superfidelity can be easily limited from above by the product w eigenvalues of the input state \(\sigma \) and the result of the erasure channel \(\xi \).

Furthermore, as shown in our examples, the quantum channel superfidelity may have potential applications in quantum control theory as an easy to compute figure of merit of quantum operations. In a simple setup, where the desired quantum channel is changed by a unitary transformation \(U_\epsilon = \exp (-\mathrm {i}\epsilon H)\) we get a linear of the decrease of channel superfidelity on the noise parameter \(\epsilon \). On the other hand, when we introduce the noise as a control error in a single qubit quantum control setup, we get a quadratic dependence on the noise parameter.

Finally, we shown numerical results for a more complicated system. We calculated the quantum channel superfidelity for a three-qubit quantum control setup. First, we found control pulses which achieve a high fidelity of the desired quantum operation, next we introduced Gaussian noise in the control pulses. Our results show that the quantum channel superfidelity stayed high for a wide range of the noise strength.

## Notes

### Acknowledgments

We would like to thank Piotr Gawron for inspiring discussions. ŁP was supported by the Polish National Science Centre under decision number DEC-2012/05/N/ST7/01105. ZP supported by the Polish National Science Centre under the post-doc programme, decision number DEC-2012/04/S/ST6/00400.

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